Graph Codes for Distributed Instant Message Collection in an Arbitrary Noisy Broadcast Network

We consider the problem of minimizing the number of broadcasts for collecting all sensor measurements at a sink node in a noisy broadcast sensor network. Focusing first on arbitrary network topologies, we provide: 1) fundamental limits on the required number of broadcasts of data gathering and 2) a general in-network computing strategy to achieve an upper bound within factor log N of the fundamental limits, where N is the number of agents in the network. Next, focusing on two example networks, namely, arbitrary geometric networks and random Erdös-Rényi networks, we provide improved in-network computing schemes that are optimal in that they attain the fundamental limits, i.e., the lower and upper bounds are tight in scaling sense. Our main techniques are three distributed encoding techniques, called graph codes, which are designed, respectively, for the above-mentioned three scenarios. Our work, thus, extends and unifies previous works such as those of Gallager and Karamchandani on the number of broadcasts for distributed function computation in special network topologies, while bringing in novel techniques, e.g., from error-control coding and noisy circuits, for both upper and lower bounds.